• Korean Journal of Mathematics
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• 제26권1호
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• pp.75-85
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• 2018

In this paper, we study nonconstant warping functions on an Einstein warped product manifold $M=B{\times}_{f^2}F$ with a warped product metric $g=g_B+f(t)^2g_F$. And we consider a 2-dimensional base manifold B with a metric $g_B=dt^2+(f^{\prime}(t))^2du^2$. As a result, we prove the following: if M is an Einstein warped product manifold with a 2-dimensional base, then there exist generally nonconstant warping functions f(t).

• 호남수학학술지
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• 제38권2호
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• pp.359-366
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• 2016

Let H be the 3-dimensional Heisenberg group, ($G=H{\times}S^1$, g) a product Riemannian manifold of Riemannian manifolds H and S with arbitrarily given left invariant Riemannian metrics respectively, and ${\Gamma}$ the discrete subgroup of G with integer entries. Then, on the Riemannian manifold ($M:=G/{\Gamma}$, ${\Pi}^*g=\bar{g}$), ${\Pi}:G{\rightarrow}G/{\Gamma}$, we evaluate the scalar curvature and the Ricci curvature.

• Kyungpook Mathematical Journal
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• 제58권2호
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• pp.347-359
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• 2018

A Kenmotsu manifold $M^n({\phi},\;{\xi},\;{\eta},\;g)$, (n = 2m + 1 > 3) is called a generalized ${\phi}-recurrent$ if its curvature tensor R satisfies $${\phi}^2(({\nabla}_wR)(X,Y)Z)=A(W)R(X,Y)Z+B(W)G(X,Y)Z$$ for all $X,\;Y,\;Z,\;W{\in}{\chi}(M)$, where ${\nabla}$ denotes the operator of covariant differentiation with respect to the metric g, i.e. ${\nabla}$ is the Riemannian connection, A, B are non-vanishing 1-forms and G is given by G(X, Y)Z = g(Y, Z)X - g(X, Z)Y. In particular, if A = 0 = B then the manifold is called a ${\phi}-symmetric$. Now, a Kenmotsu manifold $M^n({\phi},\;{\xi},\;{\eta},\;g)$, (n = 2m + 1 > 3) is said to be generalized ${\phi}-Ricci$ recurrent if it satisfies $${\phi}^2(({\nabla}_wQ)(Y))=A(X)QY+B(X)Y$$ for any vector field $X,\;Y{\in}{\chi}(M)$, where Q is the Ricci operator, i.e., g(QX, Y) = S(X, Y) for all X, Y. In this paper, we study generalized ${\phi}-recurrent$ and generalized ${\phi}-Ricci$ recurrent Kenmotsu manifolds with respect to quarter-symmetric metric connection and obtain a necessary and sufficient condition of a generalized ${\phi}-recurrent$ Kenmotsu manifold with respect to quarter symmetric metric connection to be generalized Ricci recurrent Kenmotsu manifold with respect to quarter symmetric metric connection.

• Korean Journal of Mathematics
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• 제23권2호
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• pp.313-321
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• 2015

The manifold $^*g-ESX_n$ is a generalized n-dimensional Riemannian manifold on which the differential geometric structure is imposed by the unified field tensor $^*g^{{\lambda}{\nu}}$ through the ES-connection which is both Einstein and semi-symmetric. The purpose of the present paper is to prove a necessary and sufficient condition for a unique Einstein's connection to exist in 3-dimensional $^*g-ESX_3$ and to display a surveyable tnesorial representation of 3-dimensional Einstein's connection in terms of the unified field tensor, employing the powerful recurrence relations in the first class.

• 대한수학회지
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• 제41권2호
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• pp.369-378
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• 2004

A canonical Finsler connection Nr is defined by a generalized Finsler structure called a (G, N)-structure, where G is a generalized Finsler metric and N is a nonlinear connection given in a differentiable manifold, respectively. If NT is linear, then the(G, N)-structure has a linearity in a sense and is called Berwaldian. In the present paper, we discuss what it means that NT is with a vanishing hv-torsion: ${P^{i}}\;_{jk}\;=\;0$ and introduce the notion of a stronger type for linearity of a (G, N)-structure. For important examples, we finally investigate the cases of a Finsler manifold and a Rizza manifold.

• Korean Journal of Mathematics
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• 제31권2호
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• pp.133-137
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• 2023

Consider a Riemannian manifold M equipped with a metric g. In this article, we study a notion for statistical manifolds (M, g, ∇), which can have a nonzero torsion, abbreviated to SMT. Then it turns out that the tensor fields ∇g and ${\tilde{\nabla}}g$ obtained from two different SMT-connections are different.

• 대한수학회논문집
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• 제37권2호
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• pp.585-594
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• 2022

In this article we classify four dimensional gradient Ricci solitons (M, g, f) with half harmonic Weyl curvature and at most two distinct Ricci-eigenvalues at each point. Indeed, we showed that, in a neighborhood V of each point in some open dense subset of M, (V, g) is isometric to one of the following: (i) an Einstein manifold. (ii) a domain in the Riemannian product (ℝ², g₀) × (N, ${\tilde{g}}$), where g₀ is the flat metric on ℝ² and (N, ${\tilde{g}}$) is a two dimensional Riemannian manifold of constant curvature λ ≠ 0. (iii) a domain in ℝ × W with the warped product metric $ds^2+h(s)^2{\tilde{g}}$, where ${\tilde{g}}$ is a constant curved metric on a three dimensional manifold W.

• 대한수학회논문집
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• 제38권4호
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• pp.1281-1298
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• 2023

Let (M^2m, 𝜑, g) be a B-manifold. In this paper, we introduce a new class of metric on (M^2m, 𝜑, g), obtained by a non-conformal deformation of the metric g, called a generalized Berger-type deformed metric. First we investigate the Levi-Civita connection of this metric. Secondly we characterize the Riemannian curvature, the sectional curvature and the scalar curvature. Finally, we study the proper biharmonicity of the identity map and of a curve on M with respect to a generalized Berger-type deformed metric.

• 대한수학회지
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• 제53권5호
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• pp.1149-1165
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• 2016

In this paper, we show that there exists no left invariant Riemannian metric h on the Heisenberg group H such that (H, h) is a symmetric Riemannian manifold, and there does not exist any H-invariant metric $\bar{h}$ on the Heisenberg manifold $H/{\Gamma}$ such that the Riemannian connection on ($H/{\Gamma},\bar{h}$) is a Yang-Mills connection. Moreover, we get necessary and sufficient conditions for a group homomorphism of (SU(2), g) with an arbitrarily given left invariant metric g into (H, h) with an arbitrarily given left invariant metric h to be a harmonic and an affine map, and get the totality of harmonic maps of (SU(2), g) into H with a left invariant metric, and then show the fact that any affine map of (SU(2), g) into H, equipped with a properly given left invariant metric on H, does not exist.

• Korean Journal of Mathematics
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• 제6권2호
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• pp.281-289
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• 1998

On a generalized Riemannian manifold $X_n$, we may impose a particular geometric structure by the basic tensor field $g_{\lambda\mu}$ by means of a particular connection ${\Gamma}{_\lambda}{^\nu}_{\mu}$. For example, Einstein's manifold $X_n$ is based on the Einstein's connection defined by the Einstein's equations. Many recurrent connections have been studied by many geometers, such as Datta and Singel, M. Matsumoto, and E.M. Patterson. The purpose of the present paper is to study some relations between a generalized semisymmetric $g$-recurrent manifold $GSX_n$ and its submanifold. All considerations in this present paper deal with the general case $n{\geq}2$ and all possible classes.